The purpose of the proposed psychological research is to study the ability of trainable (TMR) and educable mentally retarded children (EMR) to learn basic mathematical skills and concepts, which have usually been assumed to be unlearnable by these populations. The project will attempt to encourage and measure meaningful learning rather than conditioned responses or rote associations. The cognitive approach and analysis will focus on genuine understanding, rule-governed skills, and problem solving. Such higher order learning of basic skills and concepts is a prerequisite for further math instruction and transfer to survival skills (e.g., time-telling, making change, measuring quantities, etc.). The project will consist of a series of four TMR and four EMR training studies extending over two years' time. The focus the first year will be on counting skills (e.g., rule-governed production of the number sequence, spontaneous and appropriate application of object counting, and the abstraction of regularities about number from counting experience) and number sense (e.g., using number mentally to compare sets). Studies the second year will check basic arithmetic skills such as performing addition and subtraction mentally and with written symbols, inventing more economical procedures, etc. Of particular interest in both groups is the extent to which each is capable of abstracting such regularities as the commutativity of addition, N plus 0 equal N, and the inverse relationship of addition and subtraction. Half of the approximately 30 subjects in each study will be randomly assigned to an experimental group; half to a control group. Subjects will be matched on MA, CA, school achievement and sex. A pretest, posttest, delayed posttest design will be used. Learning will be measured rigorously in terms of transfer to new problems and everyday situations as well as in terms of long-term retention.